Metamath Proof Explorer

Theorem cdlemg12d

Description: TODO: FIX COMMENT. (Contributed by NM, 5-May-2013)

Ref Expression
Hypotheses cdlemg12.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg12.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg12.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg12.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg12b.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( ( 𝑅 ‘ 𝐹 ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg12.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg12.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemg12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg12.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemg12b.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
10 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
11 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐹 ∈ 𝑇 )
12 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐺 ∈ 𝑇 )
13 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≠ 𝑄 )
14 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) )
15 1 2 3 4 5 6 7 cdlemg12c ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) ≤ ( ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) )
16 8 9 10 11 12 13 14 15 syl133anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) ≤ ( ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) )
17 1 2 3 4 5 6 7 trlval4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) )
18 8 12 9 10 13 14 17 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) )
19 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
20 8 12 9 19 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
21 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑄 ) ≤ 𝑊 ) )
22 8 12 10 21 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑄 ) ≤ 𝑊 ) )
23 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
24 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
25 4 5 6 ltrn11at ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( 𝐺 ‘ 𝑃 ) ≠ ( 𝐺 ‘ 𝑄 ) )
26 8 12 23 24 13 25 syl113anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ≠ ( 𝐺 ‘ 𝑄 ) )
27 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) )
28 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) )
29 1 2 3 4 5 6 7 cdlemg10c ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
30 8 9 10 28 29 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
31 27 30 mtbird ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
32 1 2 3 4 5 6 7 trlval4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑄 ) ≤ 𝑊 ) ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ ( 𝐺 ‘ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) )
33 8 11 20 22 26 31 32 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) )
34 33 oveq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) = ( ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) )
35 16 18 34 3brtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( ( 𝑅 ‘ 𝐹 ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) )